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Choose an arbitrary codeword c from C. Counting the number of vectors within S$ _{t}({\bf c})$, we have c itself and all the vectors of distance 1 from c. Since c has $ n$ positions, using our definition of metric, we have that there are precisely $ n$ vectors of distance 1 from c. From Theorem 1.3, we know these spheres are mutually disjoint, therefore exactly $ \vert{\bf C}\vert(n+1)$ vectors of V are covered by the spheres of C. Thus for a 1-error correcting code, C is perfect if and only if $ \vert{\bf C}\vert(n+1)=2^{n}$, since that's how many vectors are in V.

Any perfect 1-error correcting code C $ \subset$ V has a length $ n$ of the form $ n=2^{r}-1$, where $ r$ is an integer.

Bill Cherowitzo 2001-12-11